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## G = C62.31D4order 288 = 25·32

### 15th non-split extension by C62 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.31D4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C62 — C6×C3⋊D4 — C62.31D4
 Lower central C32 — C3×C6 — C62 — C62.31D4
 Upper central C1 — C2 — C23

Generators and relations for C62.31D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a-1b3, dad=ab3, cbc-1=dbd=b-1, dcd=a3c-1 >

Subgroups: 474 in 121 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4 [×3], C22 [×3], C22 [×3], S3, C6 [×2], C6 [×12], C2×C4 [×3], D4 [×2], C23, C23, C32, Dic3 [×6], C12 [×2], D6 [×2], C2×C6 [×6], C2×C6 [×9], C22⋊C4 [×2], C2×D4, C3×S3, C3×C6, C3×C6 [×3], C2×Dic3, C2×Dic3 [×5], C3⋊D4 [×2], C2×C12 [×2], C3×D4 [×2], C22×S3, C22×C6 [×2], C22×C6 [×2], C23⋊C4, C3×Dic3 [×2], C3⋊Dic3, S3×C6 [×2], C62 [×3], C62, C6.D4, C6.D4 [×3], C3×C22⋊C4, C2×C3⋊D4, C6×D4, C6×Dic3, C6×Dic3, C3×C3⋊D4 [×2], C2×C3⋊Dic3, S3×C2×C6, C2×C62, C23.6D6, C23.7D6, C3×C6.D4, C625C4, C6×C3⋊D4, C62.31D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], Dic3 [×2], D6 [×2], C22⋊C4, C4×S3, D12, C2×Dic3, C3⋊D4 [×3], C23⋊C4, S32, D6⋊C4, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, C23.6D6, C23.7D6, D6⋊Dic3, C62.31D4

Permutation representations of C62.31D4
On 24 points - transitive group 24T581
Generators in S24
```(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 4 2 5 3 6)(7 11 9 10 8 12)(13 14 15 16 17 18)(19 24 23 22 21 20)
(1 22 5 19)(2 24 6 21)(3 20 4 23)(7 17)(8 13)(9 15)(10 14)(11 16)(12 18)
(1 17)(2 15)(3 13)(4 16)(5 14)(6 18)(7 19)(8 23)(9 21)(10 22)(11 20)(12 24)```

`G:=sub<Sym(24)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,4,2,5,3,6)(7,11,9,10,8,12)(13,14,15,16,17,18)(19,24,23,22,21,20), (1,22,5,19)(2,24,6,21)(3,20,4,23)(7,17)(8,13)(9,15)(10,14)(11,16)(12,18), (1,17)(2,15)(3,13)(4,16)(5,14)(6,18)(7,19)(8,23)(9,21)(10,22)(11,20)(12,24)>;`

`G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,4,2,5,3,6)(7,11,9,10,8,12)(13,14,15,16,17,18)(19,24,23,22,21,20), (1,22,5,19)(2,24,6,21)(3,20,4,23)(7,17)(8,13)(9,15)(10,14)(11,16)(12,18), (1,17)(2,15)(3,13)(4,16)(5,14)(6,18)(7,19)(8,23)(9,21)(10,22)(11,20)(12,24) );`

`G=PermutationGroup([(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,4,2,5,3,6),(7,11,9,10,8,12),(13,14,15,16,17,18),(19,24,23,22,21,20)], [(1,22,5,19),(2,24,6,21),(3,20,4,23),(7,17),(8,13),(9,15),(10,14),(11,16),(12,18)], [(1,17),(2,15),(3,13),(4,16),(5,14),(6,18),(7,19),(8,23),(9,21),(10,22),(11,20),(12,24)])`

`G:=TransitiveGroup(24,581);`

On 24 points - transitive group 24T614
Generators in S24
```(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 16 5 14 3 18)(2 17 6 15 4 13)(7 20 9 22 11 24)(8 21 10 23 12 19)
(2 13)(3 5)(4 17)(6 15)(7 10 22 19)(8 24 23 9)(11 12 20 21)(16 18)
(1 7)(2 23)(3 9)(4 19)(5 11)(6 21)(8 15)(10 17)(12 13)(14 22)(16 24)(18 20)```

`G:=sub<Sym(24)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,16,5,14,3,18)(2,17,6,15,4,13)(7,20,9,22,11,24)(8,21,10,23,12,19), (2,13)(3,5)(4,17)(6,15)(7,10,22,19)(8,24,23,9)(11,12,20,21)(16,18), (1,7)(2,23)(3,9)(4,19)(5,11)(6,21)(8,15)(10,17)(12,13)(14,22)(16,24)(18,20)>;`

`G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,16,5,14,3,18)(2,17,6,15,4,13)(7,20,9,22,11,24)(8,21,10,23,12,19), (2,13)(3,5)(4,17)(6,15)(7,10,22,19)(8,24,23,9)(11,12,20,21)(16,18), (1,7)(2,23)(3,9)(4,19)(5,11)(6,21)(8,15)(10,17)(12,13)(14,22)(16,24)(18,20) );`

`G=PermutationGroup([(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,16,5,14,3,18),(2,17,6,15,4,13),(7,20,9,22,11,24),(8,21,10,23,12,19)], [(2,13),(3,5),(4,17),(6,15),(7,10,22,19),(8,24,23,9),(11,12,20,21),(16,18)], [(1,7),(2,23),(3,9),(4,19),(5,11),(6,21),(8,15),(10,17),(12,13),(14,22),(16,24),(18,20)])`

`G:=TransitiveGroup(24,614);`

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 4E 6A ··· 6F 6G ··· 6Q 6R 6S 12A ··· 12F order 1 2 2 2 2 2 3 3 3 4 4 4 4 4 6 ··· 6 6 ··· 6 6 6 12 ··· 12 size 1 1 2 2 2 12 2 2 4 12 12 12 36 36 2 ··· 2 4 ··· 4 12 12 12 ··· 12

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + - - + + + + - - + image C1 C2 C2 C2 C4 C4 S3 S3 D4 Dic3 Dic3 D6 C4×S3 D12 C3⋊D4 C23⋊C4 S32 S3×Dic3 D6⋊S3 C3⋊D12 C23.6D6 C23.7D6 C62.31D4 kernel C62.31D4 C3×C6.D4 C62⋊5C4 C6×C3⋊D4 C6×Dic3 S3×C2×C6 C6.D4 C2×C3⋊D4 C62 C2×Dic3 C22×S3 C22×C6 C2×C6 C2×C6 C2×C6 C32 C23 C22 C22 C22 C3 C3 C1 # reps 1 1 1 1 2 2 1 1 2 1 1 2 2 2 6 1 1 1 1 1 2 2 4

Matrix representation of C62.31D4 in GL4(𝔽7) generated by

 5 2 3 2 1 4 3 4 2 2 3 3 0 0 0 2
,
 1 5 2 6 0 3 0 2 3 3 0 1 0 0 0 5
,
 2 0 5 2 4 2 0 4 3 3 3 1 1 6 3 0
,
 4 1 4 5 2 2 0 4 3 4 2 4 4 4 5 6
`G:=sub<GL(4,GF(7))| [5,1,2,0,2,4,2,0,3,3,3,0,2,4,3,2],[1,0,3,0,5,3,3,0,2,0,0,0,6,2,1,5],[2,4,3,1,0,2,3,6,5,0,3,3,2,4,1,0],[4,2,3,4,1,2,4,4,4,0,2,5,5,4,4,6] >;`

C62.31D4 in GAP, Magma, Sage, TeX

`C_6^2._{31}D_4`
`% in TeX`

`G:=Group("C6^2.31D4");`
`// GroupNames label`

`G:=SmallGroup(288,228);`
`// by ID`

`G=gap.SmallGroup(288,228);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,346,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d=a*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=a^3*c^-1>;`
`// generators/relations`

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